Re-export everything

content/book/preumont18_vibrat_contr_activ_struc_fourt_edition.md

@@ -347,9 +347,9 @@ If we left multiply the equation by \\(\Phi^T\\) and we use the orthogonalily re
 If \\(\Phi^T C \Phi\\) is diagonal, the **damping is said classical or normal**. In this case:
 \\[ \Phi^T C \Phi = diag(2 \xi\_i \mu\_i \omega\_i) \\]
 
-One can verify that the Rayleigh damping \ref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
+One can verify that the Rayleigh damping \eqref{eq:rayleigh\_damping} complies with this condition with modal damping ratios \\(\xi\_i = \frac{1}{2} ( \frac{\alpha}{\omega\_i} + \beta\omega\_i )\\).
 
-And we obtain decoupled modal equations \ref{eq:modal\_eom}.
+And we obtain decoupled modal equations \eqref{eq:modal\_eom}.
 
 <div class="cbox">
 
@@ -365,7 +365,7 @@ with:
 
 </div>
 
-Typical values of the modal damping ratio are summarized on table [Table 2](#table--tab:damping-ratio).
+Typical values of the modal damping ratio are summarized on [Table 2](#table--tab:damping-ratio).
 
 <a id="table--tab:damping-ratio"></a>
 <div class="table-caption">
@@ -382,15 +382,15 @@ Typical values of the modal damping ratio are summarized on table [Table 2](#tab
 
 The assumption of classical damping is often justified for light damping, but it is questionable when the damping is large.
 
-If one accepts the assumption of classical damping, the only difference between equation \ref{eq:general\_eom} and \ref{eq:modal\_eom} lies in the change of coordinates.
+If one accepts the assumption of classical damping, the only difference between equation \eqref{eq:general\_eom} and \eqref{eq:modal\_eom} lies in the change of coordinates.
 However, in physical coordinates, the number of degrees of freedom is usually very large.
-If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \ref{eq:modal\_eom} can often be restricted to theses modes.
+If a structure is excited in by a band limited excitation, its response is dominated by the modes whose natural frequencies are inside the bandwidth of the excitation and the equation \eqref{eq:modal\_eom} can often be restricted to theses modes.
 Therefore, the number of degrees of freedom contribution effectively to the response is **reduced drastically** in modal coordinates.
 
 
 #### Dynamic Flexibility Matrix {#dynamic-flexibility-matrix}
 
-If we consider the steady-state response of equation \ref{eq:general\_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
+If we consider the steady-state response of equation \eqref{eq:general\_eom} to harmonic excitation \\(f=F e^{j\omega t}\\), the response is also harmonic \\(x = Xe^{j\omega t}\\). The amplitude of \\(F\\) and \\(X\\) is related by:
 \\[ X = G(\omega) F \\]
 
 Where \\(G(\omega)\\) is called the **Dynamic flexibility Matrix**:
@@ -647,7 +647,7 @@ With:
 -   \\(T\_{me}\\) is the transduction coefficient representing the force acting on the mechanical terminals to balance the electromagnetic force induced per unit current input (in \\(N/A\\))
 -   \\(Z\_m\\) is the mechanical impedance measured when \\(i=0\\)
 
-Equation \ref{eq:gen\_trans\_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
+Equation \eqref{eq:gen\_trans\_e} shows that the voltage across the electrical terminals of any electromechanical transducer is the sum of a contribution proportional to the current applied and a contribution proportional to the velocity of the mechanical terminals.
 Thus, if \\(Z\_ei\\) can be measured and substracted from \\(e\\), a signal proportional to the velocity is obtained.
 
 To do so, the bridge circuit as shown on [Figure 15](#figure--fig:bridge-circuit) can be used.
@@ -719,7 +719,7 @@ With:
 
 #### Constitutive Relations of a Discrete Transducer {#constitutive-relations-of-a-discrete-transducer}
 
-The set of equations \ref{eq:piezo\_eq} can be written in a matrix form:
+The set of equations \eqref{eq:piezo\_eq} can be written in a matrix form:
 
 \begin{equation}\label{eq:piezo\_eq\_matrix}
 \begin{bmatrix}
@@ -769,7 +769,7 @@ It measures the efficiency of the conversion of the mechanical energy into elect
 
 </div>
 
-If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \ref{eq:piezo\_eq\_matrix\_bis} over the volume of the transducer:
+If one assumes that all the electrical and mechanical quantities are uniformly distributed in a linear transducer formed by a **stack** (see [Figure 17](#figure--fig:piezo-stack)) of \\(n\\) disks of thickness \\(t\\) and cross section \\(A\\), the global constitutive equations of the transducer are obtained by integrating \eqref{eq:piezo\_eq\_matrix\_bis} over the volume of the transducer:
 
 \begin{equation}\label{eq:piezo\_stack\_eq}
 \begin{bmatrix}
@@ -798,7 +798,7 @@ where
 
 {{< figure src="/ox-hugo/preumont18_piezo_stack.png" caption="<span class='figure-number'>Figure 17: </span>Piezoelectric linear transducer" >}}
 
-Equation \ref{eq:piezo\_stack\_eq} can be inverted to obtain
+Equation \eqref{eq:piezo\_stack\_eq} can be inverted to obtain
 
 \begin{equation}\label{eq:piezo\_stack\_eq\_inv}
 \begin{bmatrix}
@@ -830,7 +830,7 @@ The total power delivered to the transducer is the sum of electric power \\(V i\
 
 {{< figure src="/ox-hugo/preumont18_piezo_discrete.png" caption="<span class='figure-number'>Figure 18: </span>Discrete Piezoelectric Transducer" >}}
 
-By integrating equation \ref{eq:piezo\_work} and using the constitutive equations \ref{eq:piezo\_stack\_eq\_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
+By integrating equation \eqref{eq:piezo\_work} and using the constitutive equations \eqref{eq:piezo\_stack\_eq\_inv}, we obtain the analytical expression of the stored electromechanical energy for the discrete transducer:
 
 \begin{equation}\label{eq:piezo\_stack\_work}
   W\_e(\Delta, Q) = \frac{Q^2}{2 C (1 - k^2)} - \frac{n d\_{33} K\_a}{C(1-k^2)} Q\Delta + \frac{K\_a}{1-k^2}\frac{\Delta^2}{2}
@@ -882,7 +882,7 @@ And one can see that
   \frac{z^2 - p^2}{z^2} = k^2
 \end{equation}
 
-Equation \ref{eq:distance\_p\_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
+Equation \eqref{eq:distance\_p\_z} constitutes a practical way to determine the electromechanical coupling factor from the poles and zeros of the admittance measurement ([Figure 20](#figure--fig:piezo-admittance-curve)).
 
 <a id="figure--fig:piezo-admittance-curve"></a>